I guess Sparrow III refers to the missiles and not the plane. This is what Wiki says about it:

Sparrow III
Concurrently with the development of the Sparrow I, in 1951, Raytheon began work on the semi-active radar homing version of Sparrow family of missiles, the AAM-N-6 Sparrow III. The first of these weapons entered United States Navy service in 1958.
The AAM-N-6a was similar to the -6, but used a new Thiokol liquid-fuel rocket engine for improved performance. It also included changes to the guidance electronics to make it effective at higher closing speeds. The -6a was also selected to arm the Air Force's F-110A Spectre (F-4 Phantom) fighters in 1962, known to them as the AIM-101. It entered production in 1959, with 7500 being built.
Another upgrade reverted to a Rocketdyne solid-fuel motor for the AAM-N-6b, which started production in 1963. The new motor significantly increased maximum range to 35 kilometres (22 mi) for head-on attacks.
During this year the Navy and Air Force agreed on standardized naming conventions for their missiles. The Sparrows became the AIM-7 series. The original Sparrow I and aborted Sparrow II became the AIM-7A and AIM-7B, despite both being out of service. The -6, -6a and -6B became the AIM-7C, AIM-7D and AIM-7E respectively.

Reason that I'm asking is that my dad (now deceased) was an aeronautical (or aerospace) engineer in the 50's and 60's designing missiles. I have a prototype drawing of his that is of a Navy plane with FJ-404 on it and the drawing was titled, 'Sparrow III.' We found other drawings of his of missiles (being designed back then) that were first put in use many, many years later. I am going to frame the drawing of that Navy plane with its missiles and hang it in my Plebes bedroom!